16 BAIRE'S THEOREM AND ITS CONSEQUENCES 87
coefficients cn e E' are given by
Applying this result to the situation where 7 is a circuit
(Q^t<>2n) in A - {a} we obtain, for any weakly analytic function
on A, the Taylor expansion
(12.16.6.5) /,(*) = £ 1 /<»>(*)(Z - a)",
n=0 Wl
convergent in the disk \z — a\ < r (independent ofze E), and the derivatives
are given by
More generally, suppose that in A — {a} we have
(12.16.6.7) /z=^_^i_ + ^>
where the x% are linear forms on E (a priori, not necessarily continuous), and
z\~*gz is a weakly analytic function on A — {a} which is weakly bounded in a
neighborhood of a. Then, by (9.15), for each x e E, the function z\-+gs(x) can
be extended by continuity to the point a, and the extended function is analytic
in A. Hence there exists a weakly analytic function zt-*hz in A whose restric-
tion to A - {a} is z\-*gz. Furthermore, the x* are continuous linear forms on
E (i.e., they are elements of E'). Indeed, for each x e E we have
= ^ J(z - a)*
where y is a circuit t\~+a + reu (0 <J tf <i 27c) with r sufficiently small and
independent of x (9.14); our assertion therefore follows from the considera-
tions above (12.16.6.3).
Finally, we remark that the principle of analytic continuation (9.4.2)
remains valid for weakly analytic functions. This is an immediate consequence
of the definitions.
(12.16.7) Let E be a Hilbert space and u an endomorphism of the (non-
topologicaT) vector space E. Then the following three conditions are equivalent:
(a) u is continuous]
(b) u is weakly continuous•;
(c) u has an adjoint (11.5).oints (zn) in A (3.13.13),